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Properties and Operations of Fractions: Where a, b, c, and d are real numbers, variables, or algebraic expressions and b, d ≠0 𝒂𝒂 Equivalent Fractions: Example: 𝟏𝟏 𝟐𝟐 𝒃𝒃 𝟐𝟐 𝒄𝒄 = 𝒅𝒅 if and only if 𝒂𝒂𝒂𝒂 = 𝒃𝒃𝒃𝒃 = 𝟒𝟒 because (𝟏𝟏) ∗ (𝟒𝟒) = (𝟐𝟐) ∗ (𝟐𝟐) 𝒂𝒂 −𝒂𝒂 Rules of Signs: − 𝒃𝒃 = 𝒃𝒃 𝒂𝒂 −𝒂𝒂 Generate Equivalent Fractions: Example: (𝒙𝒙)(𝟑𝟑) 𝒙𝒙 = (𝟐𝟐)(𝟑𝟑) = 𝟐𝟐 𝒂𝒂 = −𝒃𝒃 and −𝒃𝒃 = 𝒃𝒃 𝟑𝟑𝟑𝟑 𝟔𝟔 𝒂𝒂 𝒂𝒂𝒂𝒂 = 𝒃𝒃𝒃𝒃 , 𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘 𝒄𝒄 ≠ 𝟎𝟎 𝒃𝒃 Add or Subtract with like denominators: Example: 𝟏𝟏 + 𝟐𝟐 𝟓𝟓𝟓𝟓 𝟐𝟐 = 𝟏𝟏+𝟓𝟓𝟓𝟓 𝟐𝟐 𝒂𝒂 𝒄𝒄 ± 𝒃𝒃 = 𝒃𝒃 Add or Subtract with unlike denominators: 𝒂𝒂 𝒄𝒄 𝒂𝒂±𝒄𝒄 𝒃𝒃 𝒂𝒂𝒂𝒂 𝒄𝒄𝒄𝒄 ± 𝒅𝒅 = 𝒃𝒃𝒃𝒃 ± 𝒃𝒃𝒃𝒃 = 𝒃𝒃 𝒂𝒂𝒂𝒂±𝒄𝒄𝒄𝒄 𝒃𝒃𝒃𝒃 (find a common denominator by multiplying top and bottom of each part by the other part’s denominator.) Example: 𝟏𝟏 𝟐𝟐 𝟑𝟑 (𝟓𝟓) 𝟏𝟏 𝒂𝒂 Multiply Fractions: Example: 𝟑𝟑 (𝟐𝟐) 𝟑𝟑 + 𝟓𝟓 = (𝟓𝟓) 𝟐𝟐 + (𝟐𝟐) 𝟓𝟓 = 𝟏𝟏 𝒄𝒄 𝒂𝒂𝒂𝒂 (𝟑𝟑)𝟏𝟏 𝒂𝒂 𝒂𝒂 (𝟓𝟓)(𝟐𝟐) = 𝟓𝟓+𝟔𝟔 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏 (multiply top times the top and bottom times the bottom) ∗ = 𝒃𝒃𝒃𝒃 𝒃𝒃 𝒅𝒅 (𝟑𝟑)(𝟏𝟏) (𝟓𝟓)(𝟏𝟏)+(𝟐𝟐)(𝟑𝟑) 𝟏𝟏 ∗ = (𝟓𝟓)(𝟑𝟑) = (𝟑𝟑)𝟓𝟓 = 𝟓𝟓 𝟓𝟓 𝟑𝟑 𝒄𝒄 𝒅𝒅 𝒂𝒂𝒂𝒂 Divide Fractions: 𝒃𝒃 ÷ 𝒅𝒅 = 𝒃𝒃 ∗ 𝒄𝒄 = 𝒃𝒃𝒃𝒃 , 𝒄𝒄 ≠ 𝟎𝟎. (if it’s division, just reciprocate the fraction you’re dividing by and change it to multiplication.) Example: 𝟑𝟑 𝟏𝟏 𝟑𝟑 𝟑𝟑 (𝟑𝟑)(𝟑𝟑) 𝟗𝟗 ÷ 𝟑𝟑 = 𝟓𝟓 ∗ 𝟏𝟏 = (𝟓𝟓)(𝟏𝟏) = 𝟓𝟓 𝟓𝟓 Trevor L.A. May 2010